Logkx

Logkx, written as log_k(x) or log_k x, denotes the logarithm of x with base k. It is defined for base k > 0 with k ≠ 1 and for x > 0, and it is the inverse of the exponential function k^t. In other words, log_k x equals the exponent to which k must be raised to obtain x. This yields log_k(k) = 1 and log_k(1) = 0. Change of base: log_k x = ln x / ln k = log_a x / log_a k for any positive a ≠ 1. Basic rules include log_k(x^n) = n log_k x, log_k(xy) = log_k x + log_k y, and log_k(1/x) = -log_k x. Monotonicity depends on k: the function is increasing for k > 1 and decreasing for 0 < k < 1. Common special cases include the natural logarithm ln x = log_e x and the common logarithm log_10 x. Examples: log_2 8 = 3, log_10 100 = 2. The graph is defined on x > 0 and tends to −∞ as x → 0+, while it tends to ∞ as x → ∞ for bases greater than 1; for 0 < k < 1 the graph is reversed.

Applications include solving exponential equations, transforming multiplicative relationships into additive ones, and working with logarithmic scales in science. See also natural logarithm, exponential function, change of base formula, and logarithmic properties.